Retkes convergence criterion

In mathematics, the Retkes convergence criterion, named after Zoltán Retkes, gives necessary and sufficient conditions for convergence of numerical series. Numerous criteria are known for testing convergence. The most famous of them is the so-called Cauchy criterion, the only one that gives necessary and sufficient conditions. Under weak restrictions the Retkes criterion gave a new necessary and sufficient condition for the convergence. The criterion will be formulated in the complex settings:

Assume that $\quad \{z_{k}\}_{k=1}^{\infty }\subset {\mathbf {C}}$ and $z_{i}\neq z_{j}\quad$ if $\quad i\neq j\quad$ . Then

\sum _{k=1}^{\infty }z_{k}=s\quad \iff \quad \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {z_{k}^{n}}{\Pi _{k}(z_{1},\ldots ,z_{n})}}=s

In the above formula $\Pi _{k}(z_{1},\ldots ,z_{n}):=(z_{k}-z_{1})(z_{k}-z_{2})\cdots (z_{k}-z_{k-1})(z_{k}-z_{k+1})\cdots (z_{k}-z_{n})\quad k=1,\ldots ,n.$

The equivalence can be proved by using the Hermite–Hadamard inequality.

References

Zoltán Retkes, "An extension of the Hermite–Hadamard Inequality", Acta Sci. Math. (Szeged), 74 (2008), pages 95–106.

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